A compact subset of $\mathbb{R}^n$ is called a Kakeya set if it contains a unit line segment in every direction. Kakeya conjecture over $\mathbb{R}^n$ predicts that the Hausdorff as well as Minkowski dimension of such sets is $n$. This is resolved for $n=1$ and $2$. Recently a resolution is proposed is for $n=3$ by Hong Wang and Joshua Zahl. Wolff formulated an analogue of Kakeya conjecture over finite fields. Several authors including Nick Katz, Terence Tao had partial results towards this conjecture before the seminal work of Zeev Dvir who settled the conjecture. In this talk, after a quick review of the known results, we will give a complete proof of the result of Zeev Dvir. Join Zoom Meeting https://zoom.us/j/95038155052 Meeting ID: 950 3815 5052 Passcode: 272075

The *minimum distance* problem (MDP) consists of finding the sparsest nonzero vector in a subspace over a finite field. We describe a reduction from the canonical NP-hard problem of solving systems of quadratic equations to the approximate version of MDP and to the related *nearest codeword problem* (the affine analog of MDP). Our reduction and proof are simple and PCP-free, based on the tensor product of a linear code whose codewords all have approximately the same weight. We hope to present the full proof in the talk. Time permitting, we will also mention analogous results for the Shortest Vector Problem on lattices. Based on joint work with Vijay Bhattiprolu, Euiwoong Lee, and Xuandi Ren.